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Home Question Bank Math Pre Calculus Archive 2024 December 11

Pre-calculus Questions from Dec 11,2024

Browse the Pre-calculus Q&A Archive for Dec 11,2024, featuring a collection of homework questions and answers from this day. Find detailed solutions to enhance your understanding.

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6) Prove using induction that for every positive integer \( n \), \( 1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\ldots+n \cdot n!=(n+1)!-1(5 \) points \( ) \) 37. Hubble Telescope The brightness, or intensity, of starlight varies inversely with the square of its distance from Earth. The Hubble Telescope can see stars whose intensities are \( \frac{1}{50} \) that of the faintest star now seen by ground-based telescopes. Determine how much farther the Hubble Telescope can see into space than ground-based telescopes. (Source: National Aeronautics and Space Administration.) Which of the following transformations are used \( f(x)=\log _{5} x \) to the graph of \( g(x)=-2 \log _{5}(x-6) \) (1 point) Shift the graph of \( f(x) 6 \) units down. Shift the graph of \( f(x) 6 \) units to the left. Stretch the graph of \( f(x) \) vertically by a factor \( c \) Stretch the graph of \( f(x) \) vertically by a factor \( c \) Which of the following transformations are used when transforming the graph of the parent function \( f(x)=\log _{7} x \) to the graph of \( g(x)=-\log _{7}(3 x)+4 \) ? Select all that apply. (1 point) Shift the graph of \( f(x) 4 \) units to the right. Reflect the graph of \( f(x) \) over the \( y \)-axis. Compress the graph of \( f(x) \) vertically by a factor of 3 . Compress the graph of \( f(x) \) units up. Incorrect: If you guessed the answer to this question, or did not answer it correctly, go back and review translations, stretches, compressions, and reflections of logarithmic functions. Which of the following transformations are used when transforming the graph of the parent function \( f(x)=\log _{7} x \) to the graph of \( g(x)=-\log _{7}(3 x)+4 \) ? Select all that apply. (1 point) Shift the graph of \( f(x) 4 \) units to the right. Reflect the graph of \( f(x) 4 \) units up. Compress the graph of \( f(x) \) vertically by a factor of 3 . Compress the graph of \( f(x) \) horizontally by a factor of \( \frac{1}{3} \). Which of the following transformations are used when transforming the graph of the parent function \( f(x)=\log _{7} x \) to the graph of \( g(x)=-\log _{7}(3 x)+4 \) ? Select all that apply. (1 point) Shift the graph of \( f(x) 4 \) units to the right. Reflect the graph of \( f(x) 4 \) units up. Compress the graph of \( f(x) \) over the \( y \)-axis. Compress the graph of \( f(x) \) horizontally by a factor of 3 . 5. Berechne zur Potenzfunktion \( f \) mit \( \begin{array}{lll}\text { (1) } f(x)=x^{4} ; & \text { (2) } f(x)=x^{5} ; & \text { (3) } f(x)=x^{6} \\ \text { die Funktionswerte: } f(3) ; f(-2) ; f(0,8) ; f(-1,5) ; f\left(-\frac{3}{2}\right) ; f(\sqrt{2})\end{array} \) Begründung: Fü die Stelle \( x \) und den Vervielfachungsfaktor \( k \) gilt für die Potenzfunktion \( f \) mit \( f(x)= \) \( f(k \cdot x)=(k \cdot x)^{-n}=k^{-n} \cdot x^{-n}=k^{-n} \cdot f(x) \) 3. Lies aus dem Graphen der Potenzfunktion mit \( y=x^{-1}\left[y=x^{-2}\right] \) ab und kontrolliere rechnerisch: \( \begin{array}{lll}\text { a) Funktionswerte an den Stellen } 0,8 ;-0,8 ; 1,3 ;-1,3 \\ \text { b) Stellen, an denen die Funktion } & \text { (1) den Wert 2, } & \text { (2) den Wert } \frac{1}{4} \text { annim }\end{array} \) 3. Find the domain and range of the relation R given by \[ R=\left\{(x, y): y=x+\frac{6}{x}, x, y \in \mathbb{N} \text { and } x<6\right\} \] 5. Die Punkte liegen auf dem Graphen der angege tionen. Bestimme die fehlende Koordinate. a) \( f(x)=\frac{1}{x^{2}} ; P_{1}(2 \mid) ; P_{2}\left(\left.-\frac{1}{2} \right\rvert\,\right) ; P_{3}\left(\left\lvert\, \frac{1}{16}\right.\right) ; P_{1} \) b) \( f(x)=\frac{1}{x^{3}} ; P_{1}(1,5 \mid) ; P_{2}\left(\left.-\frac{1}{3} \right\rvert\,\right) ; P_{3}\left(\left\lvert\,-\frac{1}{27}\right.\right) \)
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